Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting

In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a delayed predator-prey system with harvesting. We show that the positive equilibrium point is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Furthermore, using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation.


Introduction
Due to its universal existence and importance, the study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology since Lotka [1] and Volterra [2] proposed the wellknown predator-prey model [3]- [6].Recently, a new method of central manifold has been developed to study the stability of delay induced bifurcation.In this paper, we study the following system: where dot means differentiation with respect to time t , ( ) x t and ( ) y t are the prey and predator population densities, respectively.Parameter 0 r > is the specific growth rate of prey in the absence of predation and without environment limitation.K is environmental carrying capacity.The functional response of the predator is of Holling's type with , , 0 a b β > .And all parameters involved with the model are positive.The purpose of this paper is to investigate the effect of time-delay on a modified predator-prey model with harvesting.We discussed the existence of Hopf bifurcation of system (1) and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are given.

Positive Equilibrium and Locally Asymptotically Stabiliy
After some calculations, we note system (1) has no boundary equilibria.However, it is more interesting to study the dynamical behaviors of the interior equilibrium points ) The two distinct interior equilibrium points ( 1) 0 , ( ) ) We transform the interior equilibrium * * * ( , ) E x y to the origin by the transformation * x x x = − , * y y y = − .Respectively, we still denote x and y by x and y .Thus, system (1) is transformed into ( ) First, we give the condition such that * * * ( , ) E x y is locally stable.For simplicity, we denote ( ) The characteristic polynomial of ( ) where Now we consider the locally asymptotically stabiliy of the system without time-delay.Then we have holds, then it follows from the Routh-Hurwitz criterion that two roots of (6) have negative real parts.

Hopf Bifurcaion
In the section, we study whether there exists periodic solutions of system (1)  By the use of the instability result for the delayed differential Equations, in order to prove the instability of the equilibrium point, it is sufficient to show that there exists a purely imaginary iω and a positive real τ such that ( , ) 0 where ( , ) ϕ λ τ is defined in Equation (5).If iω is a root of Equation ( 7), then we have  ( ) .

The Direction and Stability of the Hopf Bifurcation
In this section, we analyze the direction and stability of the Hopf bifurcation of (3) obtained in Theorem 3 by taking τ as the bifurcation parameter.Let γ = is the Hopf bifurcation value of system (3).Rescale the time by / t t τ = to normalize the delay.The periodic solution of system (3) is equivalent to the solution of the following system ( ) ( ) ( ) We define , , i j l as nonnegative integer, define (1)   (2) , We use the method which is based on the center manifold and normal form theory, and define ( 1) 0 0 (1) In fact, we can choose (1) 100 010 001 where δ is the Dirac delta function.For where ( ) ( )  ( ) ( , ) T i q q q e ω τ θ θ = is the eigenvector of (0) A corresponding to * * iω τ .Thus, (0) ( ) A q θ = * * ( ) i q ω τ θ .From the definition of (0) (1) 100 001 010 1 (2) (2) 2 01 10 0.
Then we have In order to ensure, we need to determine the value of M , from Equation (29) we have i i i i q s q q q q d q d Mq q e f q q Mq q e f q q M q q q q q q f e q q f e Then we can choose M such as where M is the conjugate complex number of M .Next we will compute the coordinate to describe the center manifold 0 C at 0 γ = .Let t w be the solution of Equation ( 27) when 0 z and z are local coordinates for the center manifold 0 C in the direction of q and * q .Note that W is real if t w is real.We only concern with the real solutions.For solution where . . .h o t stands for higher order terms, and ( [ where ( ) From Equation (36), we have ) It follows from Equation (39) that H z z q f q q f q g z z q g z z q Comparing the coefficients of ( ) ( , ) T i q q q e ω τ θ θ = , we obtain where ( ) ( , 0) In view of Equation ( 43), we induce that when 0 θ = .
Then we have , and i ∆ is the value of the determinant i U , where i U is formed by re- placing the i th column vector of * H by another column vector .Therefore, we can determine 20 ( ) W θ and 11 ( ) W θ from Equation (51) and Equation (52).Furthermore, we can easily compute 21 g .Then the Hopf bifurcating periodic solutions of system (1) at * τ on the center manifold are determined by the following formulas  T determines the period of periodic solutions: the period increases (decreases) if 2 2 0( 0) T T > < .Therefore, we have the following results.

Conclusion
This paper introduces modified time-delay predator-prey model.Then we study the Hopf bifurcation and the stability of the system.Our results reveal the conditions on the parameters so that the periodic solutions exist surrounding the interior equilibrium point.It shows that * τ is a critical value for the time delay τ .Further- more, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated.

2 β
the Hopf-bifurcation is for- ward(backward) and the bifurcating periodic solutions exist for determines the sta- bility of the bifurcating periodic solutions.The bifurcating periodic solutions are stable (unstable) if 2

Theorem 4 . 1 Re{
The Hopf bifurcation of the system (1) occurring at * periodic solutions on the center manifold are stable (unstable) if about the interior equilibrium point Now we have the following results.